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Four-Order Superconvergent Weak Galerkin Methods for the Biharmonic Equation on Triangular Meshes

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Abstract

A stabilizer-free weak Galerkin (SFWG) finite element method was introduced and analyzed in Ye and Zhang (SIAM J. Numer. Anal. 58: 2572–2588, 2020) for the biharmonic equation, which has an ultra simple finite element formulation. This work is a continuation of our investigation of the SFWG method for the biharmonic equation. The new SFWG method is highly accurate with a convergence rate of four orders higher than the optimal order of convergence in both the energy norm and the \(L^2\) norm on triangular grids. This new method also keeps the formulation that is symmetric, positive definite, and stabilizer-free. Four-order superconvergence error estimates are proved for the corresponding SFWG finite element solutions in a discrete \(H^2\) norm. Superconvergence of four orders in the \(L^2\) norm is also derived for \(k\geqslant 3\), where k is the degree of the approximation polynomial. The postprocessing is proved to lift a \(P_k\) SFWG solution to a \(P_{k+4}\) solution elementwise which converges at the optimal order. Numerical examples are tested to verify the theories.

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Authors and Affiliations

  1. Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR, 72204, USA

    Xiu Ye

  2. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Shangyou Zhang

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  1. Xiu Ye
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  2. Shangyou Zhang
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Correspondence to Shangyou Zhang.

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The manuscript is not submitted to other journals for simultaneous consideration. The submitted work is original and is not published elsewhere in any form or language. The authors certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honorarium; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.

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Ye, X., Zhang, S. Four-Order Superconvergent Weak Galerkin Methods for the Biharmonic Equation on Triangular Meshes. Commun. Appl. Math. Comput. 5, 1323–1338 (2023). https://doi.org/10.1007/s42967-022-00201-5

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  • DOI: https://doi.org/10.1007/s42967-022-00201-5

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